This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

- Contents - Hide Contents - Home - Section 8

Previous Next

7000 7050 7100 7150 7200 7250 7300 7350 7400 7450 7500 7550 7600 7650 7700 7750 7800 7850 7900 7950

7150 - 7175 -



top of page bottom of page up down


Message: 7175

Date: Fri, 01 Aug 2003 00:23:07

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Yes, I chased this down, and found a definition here:
> 
> Yahoo groups: /tuning-math/message/5546 * [with cont.] 
> 
> Given that it is so straightforward, why do you exclude it from 
your 
> other listings?

It follows immediately from the definition.

> So, I've got as far as converting that into Python:
> 
> import math, temper
> 
> def complexity5(u, primes=None):
>    primes = primes or temper.primes[:u.maxBasis()]
>    return math.log(2) * math.sqrt(
>        primes[0]**2 * u[1,]**2 +
>        primes[0]**2 * u[1,] * u[2,] +
>        primes[1]**2 * u[2,]**2)
> 
> 
> >>If I had worked out your numbering rule, I've forgotten it now.
> >>
> >>Every time you try to explain something, you bring in more jargon 
> > 
> > terms 
> > 
> >>that I don't understand (I can't speak for anybody else).
> >>
> >>The word "metric" in particular is something that's important but 
> > 
> > you 
> > 
> >>haven't defined.
> > 
> > 
> > I've posted this before; it's standard math:
> > 
> > Metric -- from MathWorld * [with cont.] 
> 
> *That's* standard math, yes, but it doesn't say anything about 
applying 
> a metric to an exterior algebra.  And in the geometric complexity 
definiton:
> 
> Yahoo groups: /tuning-math/message/4533 * [with cont.] 
> 
> you give what is, as far as I can work out, a function of a single 
> rational number as the "metric". 

A single rational number, in terms of 5-limit note-classes, is a two-
dimensional vector. Does that explain it for you?


top of page bottom of page up down


Message: 7176

Date: Fri, 01 Aug 2003 12:02:23

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Me:
>>Given that it is so straightforward, why do you exclude it from 
> your 
>>other listings?

Gene:
> It follows immediately from the definition.

So it follows from a definition that nobody understands!  (Well does 
anybody understand it?  Speak up!)  That isn't any use at all.

Me:
>>Yahoo groups: /tuning-math/message/4533 * [with cont.] 
>>
>>you give what is, as far as I can work out, a function of a single 
>>rational number as the "metric". 

Gene:
> A single rational number, in terms of 5-limit note-classes, is a two-
> dimensional vector. Does that explain it for you?

No, that makes even less sense.  You seem to be suggesting that the two 
coefficients of an octave-equivalent 5-limit vector constitute the 
parameters for a metric.  From where I'm sitting, that's gibberish.  It 
means those two coefficients are supposed to be points in some abstract 
space.  And to be symmetric, the metric would have to treat the two 
coefficients as interchangeable, so you certainly can't multiply one by 
log(3) and the other by log(5) like you do in geometric complexity.  As 
for the g(x,x)=0 rule, that means the "distance" of 15:8 is zero, the 
same as a unison!  And even if you could somehow (using a process you 
haven't explained) get geometric complexity from such a measure, it 
would only be defined in the 5-limit.  That's a pervers way of getting a 
very simple result.

I'm guessing that "note-classes" are something to do with octave 
equivalence -- there's another piece of jargon you haven't defined.


                      Graham


top of page bottom of page up down


Message: 7178

Date: Fri, 01 Aug 2003 15:35:02

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Me:
> >>Given that it is so straightforward, why do you exclude it from 
> > your 
> >>other listings?
> 
> Gene:
> > It follows immediately from the definition.
> 
> So it follows from a definition that nobody understands!  (Well 
does 
> anybody understand it?  Speak up!)  That isn't any use at all.

Up until now I had thought you understood geometric complexity. You 
remarked once that it might be better to use logs base two for the 
definition, which is true and insightful. 

> Me:
> >>Yahoo groups: /tuning-math/message/4533 * [with cont.] 
> >>
> >>you give what is, as far as I can work out, a function of a 
single 
> >>rational number as the "metric". 
> 
> Gene:
> > A single rational number, in terms of 5-limit note-classes, is a 
two-
> > dimensional vector. Does that explain it for you?
> 
> No, that makes even less sense.  You seem to be suggesting that the 
two 
> coefficients of an octave-equivalent 5-limit vector constitute the 
> parameters for a metric.  From where I'm sitting, that's 
gibberish.  It 
> means those two coefficients are supposed to be points in some 
abstract 
> space.  

Correct.

And to be symmetric, the metric would have to treat the two 
> coefficients as interchangeable, so you certainly can't multiply 
one by 
> log(3) and the other by log(5) like you do in geometric complexity. 

Not correct. All that is required is that the matrix B(i,j) that you 
are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric matrix 
with real and positive eigenvalues.

Here's the usual grab bag from Weisstein:

Symmetric Bilinear Form -- from MathWorld * [with cont.] 

http://mathworld.wolfram.com/QuadraticForm.html * [with cont.] 

http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.] 

Hermitian Inner Product -- from MathWorld * [with cont.] 


 As 
> for the g(x,x)=0 rule, that means the "distance" of 15:8 is zero, 
the 
> same as a unison!

Eh? We are looking at g([0,0], [1, 1]) to find the distance of 15/8 
to 1. If we find the distance of 15/8 to itself, we get zero, which 
is correct.

  And even if you could somehow (using a process you 
> haven't explained) get geometric complexity from such a measure, it 
> would only be defined in the 5-limit.  That's a pervers way of 
getting a 
> very simple result.

No, because in other cases we use the metric induced on wedge 
products by the metric we started out from.

> I'm guessing that "note-classes" are something to do with octave 
> equivalence -- there's another piece of jargon you haven't defined.

I thought "note-classes" was standard music theory jargon for the 
equivalence classes defined by octave equivalence.


top of page bottom of page up down


Message: 7179

Date: Sat, 02 Aug 2003 11:04:52

Subject: Re: Graham definitions for geometric complexity and badness

From: Carl Lumma

>> No, no, Graham is a cool first name, and first names
>> should be used when possible.  :)
>
>What's wrong with just calling it geometric complexity? You are going 
>to get this mixed up with Graham's complexity for linear temperaments.

It was you who called it that!  But you're right.

-Carl


top of page bottom of page up down


Message: 7180

Date: Sat, 2 Aug 2003 11:14:27

Subject: Re: Graham definitions for geometric complexity and badnesss

From: monz@xxxxxxxxx.xxx

hi Gene,


> From: Carl Lumma [mailto:ekin@xxxxx.xxxx
> Sent: Saturday, August 02, 2003 11:05 AM
> To: tuning-math@xxxxxxxxxxx.xxx
> Subject: Re: [tuning-math] Re: Graham definitions for geometric
> complexity and badness
> 
> 
> >> No, no, Graham is a cool first name, and first names
> >> should be used when possible.  :)
> >
> > What's wrong with just calling it geometric complexity?
> > You are going to get this mixed up with Graham's complexity
> > for linear temperaments.
> 
> It was you who called it that!  But you're right.



OK, i'll make a Dictionary entry for "geometric complexity".

if you give me an opening paragraph describing what it is,
i can just use the rest of your previous post for the definition.


-monz


top of page bottom of page up down


Message: 7181

Date: Sat, 02 Aug 2003 19:28:02

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>>consistency is only defined for ets.
> >> 
> >>What Gene said about regular temperaments sounded
> >>equivalent to me.
> >
> >a definition of consistency for regular temperaments?
> >doesn't seem possible, as they're infinite, so you can
> >always find better and better approximations to anything.
> 
> Yahoo groups: /tuning-math/message/3330 * [with cont.] 

umm . . . so?


top of page bottom of page up down


Message: 7182

Date: Sat, 02 Aug 2003 22:52:34

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> >>>consistency is only defined for ets.
>> >> 
>> >>What Gene said about regular temperaments sounded
>> >>equivalent to me.
>> >
>> >a definition of consistency for regular temperaments?
>> >doesn't seem possible, as they're infinite, so you can
>> >always find better and better approximations to anything.
>> 
>> Yahoo groups: /tuning-math/message/3330 * [with cont.] 
>
>umm . . . so?

So, doesn't sound like you can break consistency in
regular temperaments!

-Carl


top of page bottom of page up down


Message: 7183

Date: Sat, 02 Aug 2003 00:45:00

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Gene Ward Smith wrote:

> Up until now I had thought you understood geometric complexity. You 
> remarked once that it might be better to use logs base two for the 
> definition, which is true and insightful. 

No, I said I didn't understand it.  I could tell from the examples that 
it involves logs of primes, and that any logarithms should be to base 2 
in this context.

In much the same way, I can see that the units depend on the 
codimension? of the wedgie.  So an interval has complexity measured in 
octaves, but a linear temperament (or pair of commas) is in octaves^2. 
Would it make sense to take root so that everything's in octaves?


> Not correct. All that is required is that the matrix B(i,j) that you 
> are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric matrix 
> with real and positive eigenvalues.

I've found that matrix!  It's

(2g1  g1  g1 ...)
( g1 2g2  g2 ...)
( g1  g2 2g3 ...)
(... ... ... ...)

where gi is the square of the logarithm of the (i-1)th prime number. 
The octave specific equivalent will have a zero eigenvalue.  Is that a 
problem?

Here's the general function for intervals.  I still don't know how to do 
more complex wedgies.


def intervalComplexity(u, primes=None):
   """
       Geometric complexity for any interval u,
       expressed as a wedgie
   """
   maxBasis = u.maxBasis()
   primes = primes or temper.primes[:maxBasis]
   result = 0.0
   for i in range(maxBasis):
       for j in range(i, maxBasis):
           result += primes[i]**2 * u[i+1,]*u[j+1,]
   # back two levels of indentation
   return math.log(2) * math.sqrt(result)


> Here's the usual grab bag from Weisstein:
> 
> Symmetric Bilinear Form -- from MathWorld * [with cont.] 

I'm not sure the point of that.

> http://mathworld.wolfram.com/QuadraticForm.html * [with cont.] 

Yes, that looks familiar, and you did mention it in the original message.

> http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.] 

And that's a special case of the above.

> Hermitian Inner Product -- from MathWorld * [with cont.] 

Indeed, I thought it looked more like an inner product than a metric. 
We can ignore the complex bit, can't we?

> Eh? We are looking at g([0,0], [1, 1]) to find the distance of 15/8 
> to 1. If we find the distance of 15/8 to itself, we get zero, which 
> is correct.

Hey, that's the first time you've mentioned that the metric is from the 
origin to the point!  That's one of the steps that's missing from the 
original definition.

>>I'm guessing that "note-classes" are something to do with octave 
>>equivalence -- there's another piece of jargon you haven't defined.
> 
> I thought "note-classes" was standard music theory jargon for the 
> equivalence classes defined by octave equivalence.

It's always best to define things that aren't common currency here, in 
case somebody doesn't know or there's an ambiguity with some other 
context.  I've heard of "pitch classes" which are octave equivalent, but 
also ignore fine tuning and so work like equal temperaments.  I think 
that makes them "finite cyclic groups".  Agmon doesn't seem to call his 
integer pairs anything.


                     Graham


top of page bottom of page up down


Message: 7184

Date: Sat, 02 Aug 2003 00:48:11

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> In much the same way, I can see that the units depend on the 
> codimension? of the wedgie.  So an interval has complexity measured 
in 
> octaves, but a linear temperament (or pair of commas) is in 
octaves^2. 
> Would it make sense to take root so that everything's in octaves?

You could if you wished, I suppose. 

> 
> > Not correct. All that is required is that the matrix B(i,j) that 
you 
> > are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric 
matrix 
> > with real and positive eigenvalues.
> 
> I've found that matrix!  It's
> 
> (2g1  g1  g1 ...)
> ( g1 2g2  g2 ...)
> ( g1  g2 2g3 ...)
> (... ... ... ...)
> 
> where gi is the square of the logarithm of the (i-1)th prime 
number. 

I think I used half of that matrix.

> The octave specific equivalent will have a zero eigenvalue.  Is 
that a 
> problem?

I don't think so--we are ignoring 2. That should make you happy. :)

> Here's the general function for intervals.  I still don't know how 
to do 
> more complex wedgies.

You can either convert to an orthonormal basis, or use my worked-out 
formulas.

> > Here's the usual grab bag from Weisstein:
> > 
> > Symmetric Bilinear Form -- from MathWorld * [with cont.] 
> 
> I'm not sure the point of that.

This gives the inner product you asked about below.

> > http://mathworld.wolfram.com/QuadraticForm.html * [with cont.] 
> 
> Yes, that looks familiar, and you did mention it in the original 
message.
> 
> > http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.] 
> 
> And that's a special case of the above.
> 
> > Hermitian Inner Product -- from MathWorld * [with cont.] 
> 
> Indeed, I thought it looked more like an inner product than a 
metric. 
> We can ignore the complex bit, can't we?

Since everything in sight is real, yes. It makes things that much 
easier.

> > Eh? We are looking at g([0,0], [1, 1]) to find the distance of 
15/8 
> > to 1. If we find the distance of 15/8 to itself, we get zero, 
which 
> > is correct.
> 
> Hey, that's the first time you've mentioned that the metric is from 
the 
> origin to the point!  That's one of the steps that's missing from 
the 
> original definition.

That's the norm, or size of the interval, which is where I started 
from (the L(p/q) function.) You can formulate things in terms of 
Euclidean metrics, Euclidean norms, inner products, bilinear forms, 
or quadradic forms, and pass from one to another. It's all closely 
related.


top of page bottom of page up down


Message: 7185

Date: Sat, 02 Aug 2003 03:11:52

Subject: Graham definitions for geometric complexity and badness

From: Gene Ward Smith

I gave a definition of geometric complexity using natural logs; 
Graham suggested log base 2 instead. He also proposed taking the dth 
root of this, where d is the codimension of the wedgie--that is, the 
number of commas used to define it.

If G is the Graham geometric complexity under this definition, R is 
(rms or minimax, etc.) error, and n = pi(p) is the number of primes 
in the p-limit we are looking at, the formula for geometic badness 
now becomes

B = R G^n

which is pretty nice. Does anyone have a concern about switching 
definitions to the Graham version, which looks to me like a good idea?


top of page bottom of page up down


Message: 7186

Date: Sat, 2 Aug 2003 01:39:08

Subject: Re: Graham definitions for geometric complexity and badness

From: monz@xxxxxxxxx.xxx

i suggest we use the name "Breed complexity".
feedback please.


-monz


> -----Original Message-----
> From: Gene Ward Smith [mailto:gwsmith@xxxxx.xxxx
> Sent: Friday, August 01, 2003 8:12 PM
> To: tuning-math@xxxxxxxxxxx.xxx
> Subject: [tuning-math] Graham definitions for geometric complexity and
> badness
> 
> 
> I gave a definition of geometric complexity using natural logs; 
> Graham suggested log base 2 instead. He also proposed taking the dth 
> root of this, where d is the codimension of the wedgie--that is, the 
> number of commas used to define it.
> 
> If G is the Graham geometric complexity under this definition, R is 
> (rms or minimax, etc.) error, and n = pi(p) is the number of primes 
> in the p-limit we are looking at, the formula for geometic badness 
> now becomes
> 
> B = R G^n
> 
> which is pretty nice. Does anyone have a concern about switching 
> definitions to the Graham version, which looks to me like a good idea?


top of page bottom of page up down


Message: 7187

Date: Sat, 02 Aug 2003 01:39:50

Subject: Re: Graham definitions for geometric complexity and badness

From: Carl Lumma

>i suggest we use the name "Breed complexity".
>feedback please.

No, no, Graham is a cool first name, and first names
should be used when possible.  :)

-C.


top of page bottom of page up down


Message: 7188

Date: Sat, 02 Aug 2003 11:45:11

Subject: Re: Graham definitions for geometric complexity and badness

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >i suggest we use the name "Breed complexity".
> >feedback please.
> 
> No, no, Graham is a cool first name, and first names
> should be used when possible.  :)

What's wrong with just calling it geometric complexity? You are going 
to get this mixed up with Graham's complexity for linear temperaments.


top of page bottom of page up down


Message: 7189

Date: Sun, 03 Aug 2003 17:43:21

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >> >>>consistency is only defined for ets.
> >> >> 
> >> >>What Gene said about regular temperaments sounded
> >> >>equivalent to me.
> >> >
> >> >a definition of consistency for regular temperaments?
> >> >doesn't seem possible, as they're infinite, so you can
> >> >always find better and better approximations to anything.
> >> 
> >> Yahoo groups: /tuning-math/message/3330 * [with cont.] 
> >
> >umm . . . so?
> 
> So, doesn't sound like you can break consistency in
> regular temperaments!

how can you break something that isn't even defined?


top of page bottom of page up down


Message: 7190

Date: Sun, 03 Aug 2003 17:53:54

Subject: Re: review requested

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> --> Whoops, wrong list!
> 
> Hey everybody,
> 
> I wrote this in a hurry, trying to explain the activity on this list
> to a friend is as short a document as possible.

i just annotated it with a bunch of corrections, but then i hit the 
backspace key (which usually acts as "delete") and was sent back to 
the previous webpage. AAARRRRRGGGGHHHH!!

> I haven't read *The Forms of Tonality* since early 2001, but I plan
> to do that again now.  I'm sure it covers much of the same ground.

not enough of it. you're getting closer to my whole philosophy on 
these things.

> I wonder what everyone thinks of this?  Anything you disagree with?
> Errors?  Just not worth fixing?

let me try again, and hopefully it won't all disappear . . .


top of page bottom of page up down


Message: 7191

Date: Sun, 03 Aug 2003 11:05:54

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> So, doesn't sound like you can break consistency in
>> regular temperaments!
>
>how can you break something that isn't even defined?

You can't!  Which is why I was said there isn't any
in linear temperaments.

You mentioned something about infinite numbers of notes
always yielding better approximations.  What did you
mean by that?

-Carl


top of page bottom of page up down


Message: 7192

Date: Sun, 03 Aug 2003 18:06:16

Subject: Re: review requested

From: Paul Erlich

is the diatonic major second just or near-just?

> () Eventually, we will run into pitches that are very close to 
pitches
> we already have.  The small intervals between such pairs of pitches
> called commas.
> 
> 	() We create a "pun" if we use the same name ("Ab") for both
> 	notes in such a pair.

i think "pun" refers to the same pitch being used for different 
musical functions.
> 
> () We can temper the comma(s) out!
> 
> 	() Doing so collapses the lattice into a finite "block".

or an infinite "strip", "slice", etc., depending on how many 
independent commas we temper out.

>  The
> 	block tiles the lattice.  To move between tiles, simply
> 	transpose all the notes in the basic block by some number of
> 	commas.

if you're tempered the relevant comma out, no transposition is 
involved at all.

> 		() You can think of "simple" as giving more intervals
> 		with fewer tones if the comma is tempered out.

??

> 
> 		() As a matter of strange coincidence, the same math 
is
> 		behind harmonic entropy!

behind or in front of?

> However, while diminished makes a good showing, you can see that
> "porcupine" is better.  Note...
> 
> 	() Diminished, but not porcupine, is to be found in 12-et.
> 
> 	() Diminished was not used by composers until after 12-et had
> 	become entrenched.
> 
> This suggests that porcupine is a potentially fertile direction for 
new
> music.  Indeed, the temperament is named after a rather fetching 
piece
> by composer Herman Miller, the "Mizarian Porcupine Overture"...
> 
>  * [with cont.]  (Wayb.)
> 
> Porcupine temperament * [with cont.]  (Wayb.)

my piece "glassic" is even more directly based on this temperament, 
using its 7-note MOS for long stretches, and was just rebroadcast on 
wnyc!

> 
> () Reactions...
> 
> >>() We can temper the comma(s) out!
> >>        () Doing so collapses the lattice into a finite "block".
> >>         The block tiles the lattice.
> >
> >There you've lost me.  I would think that when the lattice 
collapsed,
> >that would be the end of it.  What is the nature of the lattice,
> >once you've tempered the commas out?
> 
> It collapses to a regular tiling.

or, perhaps better, you could picture it as being wrapped into a 
cylinder (in the case of the infinite strip), or a torus (in the case 
of the finite block), etc.


top of page bottom of page up down


Message: 7193

Date: Sun, 03 Aug 2003 00:33:09

Subject: review requested

From: Carl Lumma

--> Whoops, wrong list!

Hey everybody,

I wrote this in a hurry, trying to explain the activity on this list
to a friend is as short a document as possible.

I haven't read *The Forms of Tonality* since early 2001, but I plan
to do that again now.  I'm sure it covers much of the same ground.

I wonder what everyone thinks of this?  Anything you disagree with?
Errors?  Just not worth fixing?

Thanks,

-Carl

________________________________________________________________________

() Assume...

	() Music involves repetition.  Sometimes, instead of an exact
	repetition, a few things are changed while the rest stay the
	same.

		() A theme played in a different mode keeps generic
		intervals (3rds, etc.) the same while pitches change
		[only true for Rothenberg-proper scales].

		() A theme played in a different key keeps absolute
		intervals the same while pitches change [as goes
		Rothenberg-efficiency, one relies more on something like
		the rules of tonal music to make the key-changes
		recognizeable].

	() Possible intervals between notes are to be taken from some
	fixed set of just or near-just intervals [to exploit the signal-
	processing capabilities of the hearing system to deliver
	information to the listener].

() So, to build a scale, we take our chosen just intervals and *stack*
them.  This generates a lattice.  In the 3-limit we get a chain.  In the
5-limit we get a planar lattice.  7-limit, we can use the face-centered
cubic lattice.  etc.

() Eventually, we will run into pitches that are very close to pitches
we already have.  The small intervals between such pairs of pitches
called commas.

	() We create a "pun" if we use the same name ("Ab") for both
	notes in such a pair.

	() We create a "comma pump" by writing a chord progression whose
	starting and ending "tonic" involve a "pun".  Every time the
	chord progression is repeated, our pitch standard moves by the
	comma involved.  Or...

() We can temper the comma(s) out!

	() Doing so collapses the lattice into a finite "block".  The
	block tiles the lattice.  To move between tiles, simply
	transpose all the notes in the basic block by some number of
	commas.

() It appears that most scales that have been popular around the world
and throughout history correspond fairly well to temperaments where very
"simple" commas have been tempered out.

	() "Simple" is measured by the distance on the lattice between
	the two pitches that generate the comma.  Thus, simple commas
	naturally tend to define smaller (fewer pitches) blocks.

		() You can think of "simple" as giving more intervals
		with fewer tones if the comma is tempered out.

() To further rate temperaments, simple may be balanced against error.
The error is determined by the (log) size of the comma  and the number
of notes in the block over which it must be distributed (which simple
already tells us).

	() That is, it's easy to find small ratios with arbitrarily
	large denominators (1001:1000).  We want temperaments based on
	the simplest ratios in a given size range.

		() As a matter of strange coincidence, the same math is
		behind harmonic entropy!

() To see a database of 5-limit temperaments database, go to...

Yahoo groups: /tuning/database/ * [with cont.] 

...or try the Excel version at...

 * [with cont.]  (Wayb.)

...Try sorting by denominator (the Excel version should be by default).
You can see that 81:80 is one of the simplest commas, and is by far the
smallest comma among the few most simple (the list itself is the result
of searching 5-limit ratio space for low size*denominator ratios).

Another comma that looks good is the major diesis (648:625), which leads
to the "diminished" temperament.  If we take 8 tones/octave of this
temperament, we get the octatonic scale of Stravinsky and Messiaen.

However, while diminished makes a good showing, you can see that
"porcupine" is better.  Note...

	() Diminished, but not porcupine, is to be found in 12-et.

	() Diminished was not used by composers until after 12-et had
	become entrenched.

This suggests that porcupine is a potentially fertile direction for new
music.  Indeed, the temperament is named after a rather fetching piece
by composer Herman Miller, the "Mizarian Porcupine Overture"...

 * [with cont.]  (Wayb.)

Porcupine temperament * [with cont.]  (Wayb.)

() Reactions...

>>() We can temper the comma(s) out!
>>        () Doing so collapses the lattice into a finite "block".
>>         The block tiles the lattice.
>
>There you've lost me.  I would think that when the lattice collapsed,
>that would be the end of it.  What is the nature of the lattice,
>once you've tempered the commas out?

It collapses to a regular tiling.  Here...

 * [with cont.]  (Wayb.)

...the "unison vectors" (Fokker's term for commas) 81:80 and 32:25
define the tile shown in red, which defines the scale shown in green.

>Is it tempered or untempered (infinite or finite)?

Blocks may be left untempered.  In the 5-limit, if you temper one comma
out you get a linear temperament (chain).  Temper both commas out and
you get an equal temperament.  In the 7-limit, 3 commas are required to
tile the lattice.  Temper one out and get a planar temperament, two out
get a linear temperament, three out get an equal temperament.

In the 3-limit, the diatonic scale may be seen as a block defined by the
2187:2048 (apotome), not tempered out.  In the 5-limit, it may be
defined by 81:80 and 25:24 with the 81:80 tempered out and the 25:24 not
tempered out.

>If it's tempered, aren't all instances the same?  If it's untempered,
>what does it mean to "tile the lattice" with a tempered block?

If all commas are tempered out, you can move from tile to tile without
changing any pitches.  For some untempered comma, you must transpose all
the pitches in your tile by it n times to get the pitches for the nth
tile away in that comma's direction (which is why Fokker called it a
vector, though Gene says it's not really a vector).

-Carl


top of page bottom of page up down


Message: 7194

Date: Sun, 03 Aug 2003 11:08:03

Subject: Re: review requested

From: Carl Lumma

>> I wrote this in a hurry, trying to explain the activity on this list
>> to a friend is as short a document as possible.
>
>i just annotated it with a bunch of corrections, but then i hit the 
>backspace key (which usually acts as "delete") and was sent back to 
>the previous webpage. AAARRRRRGGGGHHHH!!

Sorry that happened.  Thanks for trying!

>> I haven't read *The Forms of Tonality* since early 2001, but I plan
>> to do that again now.  I'm sure it covers much of the same ground.
>
>not enough of it. you're getting closer to my whole philosophy on 
>these things.

I don't think I've changed my position much.  What I have is basically
from your early posts on this topic.

-Carl


top of page bottom of page up down


Message: 7195

Date: Sun, 03 Aug 2003 11:24:19

Subject: Re: review requested

From: Carl Lumma

>is the diatonic major second just or near-just?

Not sure where this is pointed.

>> 	() We create a "pun" if we use the same name ("Ab") for both
>> 	notes in such a pair.
>
>i think "pun" refers to the same pitch being used for different 
>musical functions.

It was my understanding that it was your assumption that two Ab
notes in a score refer to the same pitch, and thus, imply meantone
temperament for common practice music.  In which case my phrasing
above is ok.  If we don't want to make that assumption I should
change it.

>> () We can temper the comma(s) out!
>> 
>> 	() Doing so collapses the lattice into a finite "block".
>
>or an infinite "strip", "slice", etc., depending on how many 
>independent commas we temper out.
>
>>  The
>> 	block tiles the lattice.  To move between tiles, simply
>> 	transpose all the notes in the basic block by some number of
>> 	commas.
>
>if you're tempered the relevant comma out, no transposition is 
>involved at all.

Yes, it seems these do not belong as sub-items to tempering it
out!  Obviously, I'm talking about the untempered case.  Huge
oversight there.

>> 		() You can think of "simple" as giving more intervals
>> 		with fewer tones if the comma is tempered out.
>
>??

Simple commas tend to define small blocks, so if all commas are
tempered out we get all the intervals with fewer notes.  Even though
a linear temp. has infinitely many notes, there must be something
similar going on...

>> 		() As a matter of strange coincidence, the same math
>>             is behind harmonic entropy!
>
>behind or in front of?

?

Anyway, this clearly doesn't belong in the doc.  But if you could
write a blurb on this for monz or someone to post, I think it'd
be interesting.

>>  * [with cont.]  (Wayb.)
>> 
>> Porcupine temperament * [with cont.]  (Wayb.)
>
>my piece "glassic" is even more directly based on this temperament, 
>using its 7-note MOS for long stretches, and was just rebroadcast on 
>wnyc!

Nice.  Do you have a link?  If I ever decide to publish this, it
will be as a web page with inline graphics, and I'll ask Herman
for a link to MPO.  Or, I'm happy to provide links at lumma.org.

>>I haven't read *The Forms of Tonality* since early 2001, but I plan
>>to do that again now.  I'm sure it covers much of the same ground.
>
>not enough of it.

I assume you mean I'm not covering enough of your ground?  My goal
is to make a document much shorter than TFOT.  Actually, maybe I
haven't even do so.  TFOT was pretty short IIRC!

-Carl


top of page bottom of page up down


Message: 7196

Date: Sun, 3 Aug 2003 12:19:18

Subject: Re: review requested

From: monz@xxxxxxxxx.xxx

hi Carl and paul,


> From: Carl Lumma [mailto:ekin@xxxxx.xxxx
> Sent: Sunday, August 03, 2003 11:24 AM
> To: tuning-math@xxxxxxxxxxx.xxx
> Subject: Re: [tuning-math] Re: review requested
> 
> 
> <snip>
>
> >> () We can temper the comma(s) out!
> >> 
> >> 	() Doing so collapses the lattice into a finite "block".
> >
> >or an infinite "strip", "slice", etc., depending on how many 
> >independent commas we temper out.
> >
> >>  The
> >> 	block tiles the lattice.  To move between tiles, simply
> >> 	transpose all the notes in the basic block by some number of
> >> 	commas.
> >
> >if you're tempered the relevant comma out, no transposition is 
> >involved at all.
> 
> Yes, it seems these do not belong as sub-items to tempering it
> out!  Obviously, I'm talking about the untempered case.  Huge
> oversight there.
> 
> >> 		() You can think of "simple" as giving more intervals
> >> 		with fewer tones if the comma is tempered out.
> >
> >??
> 
> Simple commas tend to define small blocks, so if all commas are
> tempered out we get all the intervals with fewer notes.  Even though
> a linear temp. has infinitely many notes, there must be something
> similar going on...
>
> <snip>



Carl, it's simply a matter of differing dimensions.

i think i must be misunderstanding this disucssion,
because i'd think that *you* would get this.  if i'm
having to explain this to you, then i must be missing
something.

finity does not necessarily imply a dualistic distinction
between finite and infinite.  assuming that each prime-factor
is a unique dimension in a multi-dimensional array
characterizing the mathematics of the harmony, as
successive commas are tempered out or ignored, there is
a subsequent successive reduction of the dimensions
by elimination, one dimension at a time for each comma.

... but *you* know that already, no?  so then what
what are you referring to by "something similar going on"?



-monz


top of page bottom of page up down


Message: 7197

Date: Sun, 03 Aug 2003 21:31:37

Subject: Re: review requested

From: Carl Lumma

Heya monz,

>>>>		() You can think of "simple" as giving more
>>>>		intervals with fewer tones if the comma is
>>>>		tempered out.
>>>
>>>??
>>
>>Simple commas tend to define small blocks, so if all commas are
>>tempered out we get all the intervals with fewer notes.  Even
>>though a linear temp. has infinitely many notes, there must be
>>something similar going on...
>
>Carl, it's simply a matter of differing dimensions.
>
>i think i must be misunderstanding this discussion,

I'm not sure which part of the above quote you're referring to.
The last part there is a aggressive abstraction of Paul's
complexity heuristic, which may not be accurate.

>finity does not necessarily imply a dualistic distinction
>between finite and infinite.  assuming that each prime-factor
>is a unique dimension in a multi-dimensional array
>characterizing the mathematics of the harmony, as
>successive commas are tempered out or ignored, there is
>a subsequent successive reduction of the dimensions
>by elimination, one dimension at a time for each comma.
>
>... but *you* know that already, no?  so then what
>what are you referring to by "something similar going on"?

That part refers to the simple thing.  The simpler the comma,
the lower the complexity of the resulting temperament(s).
That comes from Paul's heuristic.  Complexity can be defined
as intervals/notes.

-Carl


top of page bottom of page up down


Message: 7198

Date: Sun, 03 Aug 2003 22:33:21

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>consistency is only defined when each just interval has a
>best approximation in the tuning. with an infinite number
>of irrational notes, there is no best approximation, you
>can keep finding better and better ones.

But you have to respect the map, according to Gene.  So
we can still define something like best approx. of n/p
must be best approx. n - best approx p.

-Carl


top of page bottom of page up down


Message: 7199

Date: Sun, 03 Aug 2003 22:36:13

Subject: Re: review requested

From: Carl Lumma

>>>not enough of it. you're getting closer to my whole philosophy on 
>>>these things.
>> 
>> I don't think I've changed my position much.
>
>your position? i mean you're getting closer than the forms of
>tonality alone.

Oh, thanks.  I thought you meant I needed to get closer to it.
Then maybe it's worth getting everybody's Seal of Approval on
the present doc and publishing it on the web, with inline
graphics.  I'll need a link to glassic.  If you haven't changed
the file since the mp3.com days, I have it, and can provide a
link for you.

-Carl


top of page bottom of page up

Previous Next

7000 7050 7100 7150 7200 7250 7300 7350 7400 7450 7500 7550 7600 7650 7700 7750 7800 7850 7900 7950

7150 - 7175 -

top of page